An algebra of pseudodifferential operators
type: Item
instance of: scholarly article
title: An algebra of pseudodifferential operators
zbMATH Open document ID: 0840.35130
published in: Mathematical Research Letters
publication date: 1995-11-16T00:00:00Z
review text: The author considers pseudo-differential operators in \(\mathbb{R}^n\) with symbol \(a(z)\), \(z= (x, \xi)\), satisfying the following property: \[ \sup_j |(f_j a)\hat{}(\zeta)|\in L^1(\mathbb{R}^{2n}_\zeta), \] where \(j\) runs over the lattice of the \(2n\)-tuples of integers, and \[ f_j(z)= f_0(z- j)\in C^\infty_0(\mathbb{R}^{2n}) \] with \(\sum_j f_j= 1\). Such symbols turn out to be bounded continuous functions in \(\mathbb{R}^{2n}\). The author proves that the corresponding class of pseudo-differential operators: 1) is independent of the \(t\)-quantization, \(t\in \mathbb{R}\), \[ \text{Op}_t(a) u(x)= (2\pi)^{- n} \int e^{i(x- y)\xi} a(tx+ (1- t) y,\xi) u(y) dy d\xi; \] 2) is stable under composition; 3) consists of \(L^2\)-bounded operators.
reviewed by: Luigi Rodino
Mathematics Subject Classification ID: 35S05
zbMATH DE Number: 817686
zbMATH Keywords: symbols with special properties
zbMATH Keywords: bounded continuous functions
author: Johannes Sjöstrand
MaRDI profile type: Publication
DOI: 10.4310/MRL.1994.V1.N2.A6
description: scientific article; zbMATH DE number 817686